Question: Multiply and simplify the following complex numbers: $({1+2i}) \cdot ({1-4i})$
Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({1+2i}) \cdot ({1-4i}) = $ $ ({1} \cdot {1}) + ({1} \cdot {-4i}) + ({2i} \cdot {1}) + ({2i} \cdot {-4i}) $ Then simplify the terms: $ (1) + (-4i) + (2i) + (-8i^2) $ Imaginary unit multiples can be grouped together. $ 1 + (-4 + 2)i - 8 i^2 $ After we plug in $i^2 = -1$, the result becomes $ 1 + (-4 + 2)i - (-8) $ The result is simplified: $ (1 + 8) + (-2i) = 9-2i $